# Relativity: Time dilation and distance calculations

• WWCY
In summary: I'm wondering if there is something like a "proper" distance...?Yes, there is a concept of proper distance in special relativity. The proper distance is the distance measured by an observer who is at rest relative to the objects being measured. In the mirror clocks thought experiment, the proper distance between the two mirrors is the distance as measured by an observer in the S' frame who is at rest relative to the mirrors. This distance is not the same as the distance measured by an observer in the S frame who is moving relative to the mirrors. This is because of the effects of time dilation and length contraction.
WWCY

## Homework Statement

Could someone help point out certain conceptual errors in my interpretation of the mirror clocks time dilation thought experiment? Say S' is the frame of reference traveling at speed V w.r.t to frame S, which means that events happen at the same coordinate in S'.

We know that the measurement of time between the light leaving the source and making a round trip back to the source in this frame is Δt0= 2d/c, where d is the vertical distance between the 2 mirrors. And that the time measured in the non-proper (is this a correct way of putting this?) frame S is Δt = γΔt0, which is in part due to the distance measured between the 2 events VΔt in this frame.

Here's where my understanding becomes messy. If i wished to calculate the distance moved by S', how do I go about doing it? Do I use VΔt? Can i use VΔt0 as well (my current understanding is that speed V is identical in both frames after sticking values into Lorentz velocity equations)? If so won't the distance traveled be shorter in the S' frame than the S frame? And how do I interpret these facts?
Thanks for helping!

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WWCY said:

## Homework Statement

Could someone help point out certain conceptual errors in my interpretation of the mirror clocks time dilation thought experiment? Say S' is the frame of reference traveling at speed V w.r.t to frame S, which means that events happen at the same coordinate in S'.

We know that the measurement of time between the light leaving the source and making a round trip back to the source in this frame is Δt0= 2d/c, where d is the vertical distance between the 2 mirrors. And that the time measured in the non-proper (is this a correct way of putting this?) frame S is Δt = γΔt0, which is in part due to the distance measured between the 2 events VΔt in this frame.

Here's where my understanding becomes messy. If i wished to calculate the distance moved by S', how do I go about doing it? Do I use VΔt? Can i use VΔt0 as well (my current understanding is that speed V is identical in both frames after sticking values into Lorentz velocity equations)? If so won't the distance traveled be shorter in the S' frame than the S frame? And how do I interpret these facts?
Thanks for helping!

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## The Attempt at a Solution

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Although S and S' agree about their relative speed, they have different results for the time between two events. In this case, the events being a tick of the light clock in S'.

Can you analyse it further?

PeroK said:
Although S and S' agree about their relative speed, they have different results for the time between two events. In this case, the events being a tick of the light clock in S'.

Can you analyse it further?
I'm not quite sure what you mean by analysing further, could you guide me along?Edit: Does the further analysis involve contraction?

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WWCY said:
I'm not quite sure what you mean by analysing further, could you guide me along?Edit: Does the further analysis involve contraction?

In the S frame: a point at rest in the S' frame moves a distance ##v \Delta t## in the time between the two events.

In the S' frame: a point at rest in the S frame moves a distance ##v \Delta t_0## in the time between the two events.

PeroK said:
In the S frame: a point at rest in the S' frame moves a distance ##v \Delta t## in the time between the two events.

In the S' frame: a point at rest in the S frame moves a distance ##v \Delta t_0## in the time between the two events.

Is it that different frames of reference perceive different lengths/distances?

WWCY said:
Is it that different frames of reference perceive different lengths/distances?

That's why I said "analyse" further. This statement too vague to analyse.

Also, note that in classical physics different frames measure different distances between two events. The distance between these two events in the S' frame is 0 (and that is true in both relativitity and classical physics); and the distance between the two events in the S frame is ##v \Delta t## (and that is also true in relativity and in classical physics).

PeroK said:
In the S frame: a point at rest in the S' frame moves a distance ##v \Delta t## in the time between the two events.

In the S' frame: a point at rest in the S frame moves a distance ##v \Delta t_0## in the time between the two events.

PeroK said:
That's why I said "analyse" further. This statement too vague to analyse.

Also, note that in classical physics different frames measure different distances between two events. The distance between these two events in the S' frame is 0 (and that is true in both relativitity and classical physics); and the distance between the two events in the S frame is ##v \Delta t## (and that is also true in relativity and in classical physics).

Apologies but I'm not sure what I'm supposed to be analysing, could you nudge me along?

WWCY said:
Apologies but I'm not sure what I'm supposed to be analysing, could you nudge me along?

You seem to have some vague idea from your OP that something is wrong. That somehow "lengths" and/or "distances" are not what they should be.

But, until you can define what you mean by a length or a distance and why it's not what it should be, there is nothing to say.

Can you give a precise question highlighting your concern?

PeroK said:
You seem to have some vague idea from your OP that something is wrong. That somehow "lengths" and/or "distances" are not what they should be.

But, until you can define what you mean by a length or a distance and why it's not what it should be, there is nothing to say.

Can you give a precise question highlighting your concern?

I'm wondering if there is something like a "proper" distance or length in the same way time would be considered proper in S' 's frame of reference. Is a measurement of length more fundamental in a particular frame of reference than others? And if i was asked to compute distance using speed and time intervals, should I be giving the distance relative to all the frames in question?

WWCY said:
I'm wondering if there is something like a "proper" distance or length in the same way time would be considered proper in S' 's frame of reference. Is a measurement of length more fundamental in a particular frame of reference than others? And if i was asked to compute distance using speed and time intervals, should I be giving the distance relative to all the frames in question?

There's a clue in classical physics. Objects have a length. In classical physics this is an invariant between reference frames. You may already know or guess that the length of an object is not frame-invariant in SR.

Next step: define the length of an object. (Although, this is not homework help, this is trying to teach you SR!)

Distances in classical physics are not (necessarily) invariant - it depends on what you mean by distance. So, in both classical physics and SR, you have to carefully define what distance you are talking about. A distance could be:

The distance between two fixed points; the distance an object has traveled in a time ##\Delta t##; the distance between two moving particles at time ##t##.

Anyway, this is the sort of analysis which you perhaps could have been able to do yourself. I.e. think about concepts like "distance" and analyse what is meant by them.

## 1) What is time dilation?

Time dilation is the phenomenon where time appears to pass slower for an object or person in motion relative to another object or person. This is due to the effects of special relativity, where the passage of time is perceived differently depending on the relative speed between observers.

## 2) How does time dilation affect distance calculations?

Time dilation affects distance calculations by altering the perceived distance between two objects in motion. Objects moving at high speeds will appear to be shorter in length, which can impact distance calculations. This can also affect the measurement of time intervals, which can be perceived differently by observers in motion relative to each other.

## 3) What is the formula for calculating time dilation?

The formula for calculating time dilation is t = t0 / √(1-v2/c2), where t is the perceived time, t0 is the proper time, v is the relative velocity, and c is the speed of light. This formula is known as the time dilation equation and is a fundamental concept in special relativity.

## 4) Can time dilation be observed in everyday life?

Yes, time dilation can be observed in everyday life, although the effects may be small. For example, GPS satellites in orbit experience time dilation, which must be accounted for in order for them to accurately function. Additionally, high-speed particles in particle accelerators also exhibit time dilation, which is crucial for experiments in particle physics.

## 5) How does time dilation relate to the concept of spacetime?

Time dilation is a fundamental aspect of the theory of special relativity, which is one of the pillars of modern physics. It is closely related to the concept of spacetime, which combines the three dimensions of space with the dimension of time to create a four-dimensional model. Time dilation is a consequence of the warping of spacetime by massive objects and the effects of relative velocity on the perception of time.

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